Question

how would I go about showing that the vector space of two pairs of vectors is the same? I have [1 1 0] and [0 1 0.5], then [2 1 -0.5] and [1 -1 -1] (all transposed)

Answer 1

find the nul spaces to compare with for starters is my first idea

Answer 2

can vector spaces be not the same if they have the same nul space?

Answer 3

i spose they can

Answer 4

if you cross each vector pair and they result in a parallel vector; then that would indicate to me that they are in the same direction and anchored at the origin

Answer 5

I can see that you can you can form either of the second two span vectors as linear combinations of the first two, but not sure how to show this shows vector space is the same (though I can see how it is) I don't really understand what you mean by nul spaces though

Answer 6

1st cross gives me <1,-1,2>

2nd cross <3,-3,6>

so they have the same "normal" vector in space

and each would then be anchored to the origin as well

same same normals from same points = same planes to me

Answer 7

that's very helpful, thank you :)

Answer 8

also:

create matrixes A and B

if (A|B) row reduces to (I|N)

and

(B|A) row reduces to (I|N)

then they should be equal

Answer 9

i like the plane idea better, im not to sure about that last idea :)